Normal function

Remarks

Example: with ordinal arithmetic, $f(\alpha) = \omega + \alpha$ is normal. It is strictly increasing, and $\omega + \lambda = \sup_{\beta < \lambda} (\omega + \beta)$ holds by the definition of addition at limits. Non-example: $g(\alpha) = \alpha + 1$ is strictly increasing but not continuous. Indeed $\sup_{n < \omega} g(n) = \omega$, while $g(\omega) = \omega + 1$. Other normal functions are $\alpha \mapsto \omega^\alpha$ and $\alpha \mapsto \aleph_\alpha$. A strictly increasing $f$ always satisfies $f(\alpha) \ge \alpha$. Fixed-point lemma: for each normal $f$ and each ordinal $\gamma$, there is an ordinal $\alpha \ge \gamma$ with $f(\alpha) = \alpha$. The fixed points form a closed unbounded class, and the function that lists them in increasing order is again normal. The same definition is used for functions from a limit ordinal to itself, or from a regular cardinal to itself.